62. If $y(t)=3 t^3+2 t^2-7 t+3$, find $\frac{d y}{d t}$ at $t=-1$.
A. -1 B. 1 C. -2 D. 2 .
63. If $y^2+x y-x=0$, find $\frac{d y}{d x}$ at $(0,2)$. A. $\frac{1}{2}$
B. $\frac{2}{3}$
C. $\frac{1}{3}$
D. $-\frac{1}{4}$.
64. Find value of $y^{\prime}$ if $x=3 t, y=t^2-4$ at $t=3$.
A. 2
B. $\frac{1}{2}$
C. -4
D. -2 .
65. If $f(x)=\frac{x+1}{x^3-1}, x \neq 1$, find $f^{\prime}(0)$. A. 2
B. 3
C. -2
D. -1 .
66. If $f(x)=\left(3+e^x\right)^2$. Then $f^{\prime \prime}(0)$.
A. 10 B. -10
C. -12
D. 12.
67. Find $y^{\prime}$ if $y=\left(\frac{x-1}{x+1}\right)^2$ at $x=0$. A. 0
B. -2
C. -3
D. -4 .
68. The derivative $\frac{d y}{d x}$ of $x^2-x y=0$ at $(1,0)$ is ..
A. $\frac{1}{2}$
B. 2
C. $\frac{1}{3}$
D. 3.
69. If $V=t^3+5 t$. Find $\dot{V}$ at $t=0$.
A. -5
B. 0
C. 5
D. 3.
70. Find the derivative of $x^2+y^2=5$.
A. $\frac{x}{y}$
B. $-\frac{x}{y}$
C. $\frac{2}{x}$
D. $-\frac{y}{x}$.
71. The derivative of $y=x \ln x$ at $x= e$ is
A. 0
B. 1
C. 2
D. 3.
72. Find $\frac{d y}{d x}$ if $y=\cos 2 t$ and $x=\sin 2 t$.
A. $-\cot 2 t$ B. $-\tan 2 t$
C. $-2 \tan 2 t$ D. $2 \cot 2 t$.
73. Find $y^{\prime \prime}(0)$ if $y=3 e^{2 x}+\sin x$.
A. 3 B. -12
C. -3
D. 12 .
74. Find $d y$ if $3 x^3+4 y^3=9$
A. $-\frac{3 x^2}{4 y^2} d x$
B. $-\frac{4 x^3}{3 y^3} d x$
C. $\frac{3 x^2}{1 y^2} d x$
D. $-\frac{9 x^2}{x^2}$. $9 x^2$.
Answer (1)
Find the derivative of y ( t ) with respect to t : d t d y = 9 t 2 + 4 t − 7 .
Substitute t = − 1 into the derivative: d t d y ∣ t = − 1 = 9 ( − 1 ) 2 + 4 ( − 1 ) − 7 .
Calculate the result: 9 − 4 − 7 = − 2 .
The derivative at t = − 1 is − 2 .
Explanation
Finding the derivative We are given the function y ( t ) = 3 t 3 + 2 t 2 − 7 t + 3 and asked to find its derivative d t d y at t = − 1 . First, we need to find the derivative of y ( t ) with respect to t . Using the power rule, we have d t d y = 9 t 2 + 4 t − 7
Evaluating at t=-1 Now, we substitute t = − 1 into the derivative: d t d y ∣ t = − 1 = 9 ( − 1 ) 2 + 4 ( − 1 ) − 7 = 9 − 4 − 7 = − 2
Final Answer Therefore, the derivative of y ( t ) at t = − 1 is -2.
Stating the answer The answer is -2.
Examples
Understanding derivatives is crucial in physics, especially when analyzing motion. For example, if y ( t ) represents the position of a particle at time t , then d t d y gives the particle's velocity. Knowing the velocity at a specific time, like t = − 1 in our problem, helps predict the particle's future trajectory or understand its past behavior. This concept extends to various fields, such as engineering, where understanding rates of change is essential for designing efficient systems.
